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Hysteresis suppression and synchronization near 3:1 subharmonic resonance
Abdelhak Fahsib, Mohamed Belhaqa,*
aUniversity Hassan II-Casablanca, Laboratory of Mechanics, Casablanca, Morocco
bUniversity Hassan II-Mohammadia, FSTM, Mohammadia, Morocco
a r t i c l e i n f o
Article history:
Accepted 26 February 2009 Communicated by Prof. G. Iovane
a b s t r a c t
Effect of a high-frequency excitation on hysteresis and synchronization area in a forced van der Pol–Duffing oscillator near the subharmonic 3:1 resonance is analyzed. By means of perturbation techniques, the 3:1 resonance area and the quasiperiodic modulation domain are captured. The results shown that adding a high-frequency excitation causes the back- bone curves to shift and to bend from softening to hardening. The results also reveal that a high-frequency excitation significantly affects the resonance and the synchronization zones. Further, jump phenomenon occurring in a certain range of the forcing frequency suggesting that hysteresis near the 3:1 subharmonic resonance cannot be ignored in prac- tical applications.
Ó2009 Elsevier Ltd. All rights reserved.
1. Introduction
In a recent work[1], synchronization and hysteresis suppression in a forced van der Pol–Duffing oscillator was studied near the primary resonance 1:1. It was shown that adding a high-frequency excitation (HFE) can suppress hysteresis for a certain range of the high-frequency. Previous works used HFE to study nontrivial effects on stability of equilibria[2], linear stiffness[3], natural frequencies[4], resonant behavior[5]and limit cycle[7,6].
The purpose of the present paper is to analyze the effect of a HFE on the synchronization area and on hysteresis in a forced van der Pol–Duffing oscillator near the subharmonic resonance 3:1. We use perturbation analysis to generate analytical expression predicting the values range of the fast excitation for which the hysteresis is eliminated.
The canonical model that captures the essential phenomena to be examined is a forced van der Pol–Duffing oscillator sub- jected to a HFE in the dimensionless form
€xþx ðabx2Þx_cx3¼hcosxtþaX2cosxcosXt ð1Þ
where dampinga,b, nonlinearitycand excitation amplitudeshandaare small. Dots denote differentiation with respect to timet. We assume that the frequencyXof the rapid excitation is large compared to the frequency of the external forcingx such that the resonance between the two frequencies cannot occur. In a previous work[8], a van der Pol–Mathieu–Duffing equation was investigated near the principal 2:1 and the subharmonic 1:1 resonances. It was shown that adding a HFE shifts the backbone curve and changes the nonlinear characteristic behavior of the system near these resonances from softening to hardening. In the present work, we focus our attention on the effect of a HFE on synchronization and hysteresis area in Eq.(1) close to the 3:1 subharmonic resonance.
The paper is organized as follows: In Section2, we use averaging method to derive a slow dynamic and we apply the mul- tiple scales method to the slow dynamic to obtain an autonomous slow flow. Analysis of equilibria of this slow flow provides
0960-0779/$ - see front matterÓ2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.chaos.2009.02.043
*Corresponding author.
E-mail address:mbelhaq@yahoo.fr(M. Belhaq).
Contents lists available atScienceDirect
Chaos, Solitons and Fractals
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c h a o s
analytical approximations of the 3:1 frequency response. In Section3, a multiple scales method is performed in a second step on the slow flow to approximate quasiperiodic solution and its modulation domain near the resonance. Analytical prediction of the variation of the hysteresis zone as function of the fast frequencyXas well as the influence of HFE on this area is pro- vided. We perform numerical simulation and we compare with the analytical finding for validation. Section4concludes the work.
2. Slow flow and synchronization
We use the method of direct partition of motion (DPM)[9]to derive the slow dynamic of system(1). We introduce two different time scales, a fast timeT0¼Xtand a slow timeT1¼t, and we split upxðtÞinto a slow partzðT1Þand a fast part
/ðT0;T1Þas follows
xðtÞ ¼zðT1Þ þ/ðT0;T1Þ ð2Þ
wherezdescribes slow main motions at time-scale of oscillations,/stands for an overlay of the fast motions andindicates that/is small compared toz. SinceXis considered as a large parameter we chooseX1, for convenience. The fast part
/and its derivatives are assumed to be 2p-periodic functions of fast timeT0with zero mean value with respect to this time, so thathxðtÞi ¼zðT1Þwherehi 21pR2p
0 ðÞdT0defines time-averaging operator over one period of the fast excitation with the slow timeT1 fixed.
Following[1,8], the equation governing the slow dynamic of(1)can be written as
€zþx20z ðabz2Þz_nz3¼hcosxt ð3Þ
wherex20¼112ðaXÞ2,n¼c13ðaXÞ2and an overdot denotes differentiation with respect to timet. In Eq.(3), it appears that the effect of the HFE introduces additional apparent stiffness in the linear and in the nonlinear stiffness. These effects have been observed in a spring-connected two-link mechanism at a vibrating support[2,5].
To analyze the effect of a HFE on synchronization and on hysteresis near the resonance 3:1, we introduce the detuning parameterraccording to
x20¼ x
3
2
þr ð4Þ
To derive a slow flow, we implement a first perturbation technique using the parameterl. According to(4), (3)is written as
€zþ x
3
2
z¼hcosxtþlfrzþ ðabz2Þz_þnz3g ð5Þ
We seek a two scale expansion[10]of the solution in the form
zðtÞ ¼z0ðT0;T1Þ þlz1ðT0;T1Þ þOðl2Þ ð6Þ
whereTi¼lit. In terms of the variablesTi, the time derivatives becomedtd¼D0þlD1þOðl2Þandd2
dt2¼D20þ2lD0D1þOðl2Þ whereDji¼ oj
ojTi. Substituting(6)into(5)and equating coefficients of the same power ofl, we obtain a set of linear partial differential equations
D20z0ðT0;T1Þ þ x
3
2
z0ðT0;T1Þ ¼hcosxt; ð7Þ
D20z1ðT0;T1Þ þ x
3
2
z1ðT0;T1Þ ¼ 2D0D1z0rz0þ ðabz20ÞD0z0þnz30 ð8Þ The solution to the first order is given by
z0ðT0;T1Þ ¼rðT1Þcos x
3T0þhðT1Þ
þFcosxt ð9Þ
Substituting(9)into(8), removing secular terms, we obtain to the first order the autonomous slow flow modulation equa- tions of amplitude and phase
dr
dt¼ArBr3 ðH1sin 3hþH2cos 3hÞr2 dh
dt¼SCr2 ðH1sin 3hH2cos 3hÞr ð10Þ
whereA¼a2bF4,B¼b8,C¼89nx,S¼23xr9nF4x2,H1¼9nF8x,H2¼bF8, and F¼89hx2.
Equilibria of the slow flow(10), corresponding to periodic oscillations of Eq.(3), are determined by settingdrdt¼dhdt¼0. We obtain the amplitude–frequency response equation
A2r4þA1r2þA0¼0 ð11Þ
1032 A. Fahsi, M. Belhaq / Chaos, Solitons and Fractals 42 (2009) 1031–1036
whereA2¼B2þC2,A1¼ ð2ABþ2SCþH21þH22ÞandA0¼A2þS2.
Fig. 1a shows the variation ofzðtÞ-amplitude–frequency response, as given by Eqs.(9) and (11)forh¼1 andX¼0; the other parameters are fixed asa¼0:01,b¼0:05,c¼0:1 anda¼0:02. The solid line denotes stable branch and the dashed line denotes unstable one. For validation, analytical approximations are compared to numerical integration (circles) using a Runge–Kutta method. The effect ofXon the amplitude–frequency response is illustrated inFig. 1b for different values ofX.
The plots in this figure indicate that asXincreases, the backbone curve shifts left and its nonlinear characteristic changes from softening to hardening. Near the thresholdXccorresponding to the vanishing of the nonlinear stiffness of the slow dy- namic(3), and given by
Xc¼ ffiffiffiffiffiffi 3c
p
a ð12Þ
the backbone curve becomes smaller, the synchronization area reduced and jumps are attenuated. Hysteresis is completely eliminated in the vicinity ofXc.
InFig. 2we show ther-amplitude–frequency response, as given by(11), near the 3:1 resonance.Fig. 2a indicates that for X¼0, the 3:1 resonance area is small. In contrast, forX¼50 (Fig. 2b) the 3:1 resonance area becomes larger indicating that in the hardening case the 3:1 resonance is significant in terms of magnitude and width. Note that inFig. 1we plotted thezðtÞ- amplitude–frequency response to compare with numerical integration at the slow dynamic level (Fig. 1a), whereas we plot- ted inFig. 2ther-amplitude–frequency response.
2.7 2.75 2.8 2.85 2.9 2.95 3
0 0.5 1 1.5 2
ω
z−amplitude
Ω=0
(a)
2.4 2.5 2.6 2.7 2.8 2.9 3
0 0.5 1 1.5 2
ω
z−amplitude
Ω=40 35 30 25 20 15 0
(b)
Fig. 1.Amplitude–frequency response of the slow dynamic zðtÞ. Analytical approximation: solid (for stable) and dashed (for unstable). Numerical simulation: circles. (a)X¼0, (b) effect of different values ofXon the backbone curve.
2 2.5 3 3.5
0 0.5 1 1.5 2 2.5 3
ω
r−amplitude
Ω =0 (a)
2 2.5 3 3.5
0 0.5 1 1.5 2 2.5 3
ω
r−amplitude
Ω =50 (b)
Fig. 2.Amplitude–frequency response near 3:1 resonance.
3. Quasi-periodic modulation and hysteresis
We construct analytical approximation of the limit cycle of the slow flow(10)corresponding to quasiperiodic motion of the slow dynamic(3)near the 3:1 subharmonic resonance. The Cartesian system corresponding to the polar form(10) is written as
du
dt ¼SvþgfAu ðBuþCvÞðu2þv2Þ þ2H1uvH2ðu2v2Þg dv
dt ¼ SuþgfAv ðBvCuÞðu2þv2Þ þH1ðu2v2Þ þ2H2uvg ð13Þ
where the bookkeeping parameter gis introduced in damping and nonlinearity terms. Using the multiple scales method [10], the solution to the first order of system(13)is given by
u0ðT0;T1Þ ¼RðT1ÞcosðmT0þuðT1ÞÞ
v0ðT0;T1Þ ¼ m
SRðT1ÞsinðmT0þuðT1ÞÞ ð14Þ
wherem¼jSjis the natural frequency of system(13)corresponding to the frequency of the slow flow limit cycle. The ampli- tudeRand the phaseuvary with time according to the following ‘‘slow slow” flow system ([11,12])
dR
dt ¼ARBR3 du
dt ¼ SC
jSjR2 ð15Þ
The approximate periodic solution of the slow flow(13)is then given by uðtÞ ¼RcosðmtþuÞ
vðtÞ ¼ mSRsinðmtþuÞ ð16Þ
and finally, the approximate quasiperiodic response of the slow dynamic(3)reads
zðtÞ ¼uðtÞcos x
3t
þvðtÞsin x
3t
þFcosxt ð17Þ
where the amplitudeRand the phaseuare obtained by settingdRdt¼0 and given respectively by R¼
ffiffiffiA B r
ð18Þ
u¼ SC jSjR2
t ð19Þ
At the first order, the modulated amplitude of the quasiperiodic oscillations is given by
rðtÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u20ðtÞ þv20ðtÞ q
¼R ð20Þ
This amplitude is plotted inFig. 3by a dashed horizontal line located exactly in the middle of the quasiperiodic modulation domain obtained numerically using Runge–Kutta method (solid lines). Note that to analytically capture this modulation do- main, as done for the primary resonant case 1:1[1], we require approximation ofu1ðtÞandv1ðtÞusing the second order per- turbation treatment which is beyond the scope of this paper. Fig. 3 shows the synchronization zone along with the quasiperiodic modulation domain of the slow flow limit cycle forX¼0 andX¼40.Fig. 3a indicates that the 3:1 synchro- nization can occur in a small range of the forcing frequencyxdelimited betweenx¼2:862 andx¼2:939. In addition, this plot shows the existence of two hysteresis loops. One hysteresis is found to be very small, whereas the other one is large and may produce a significant jump phenomenon from a large to a lower amplitude. A reverse situation is obtained in the hard- ening case (Fig. 3b).
InFig. 4we show the variation of hysteresis area as function of the frequencyXobtained from Eq.(11). This plot indicates that in the softening regionðX<XcÞthe hysteresis area is relatively small, whereas in the hardening zoneðX>XcÞthe hys- teresis area becomes large. The hysteresis phenomenon is attenuated in the vicinity the thresholdXc and completely sup- pressed atXc.
4. Conclusion
In this paper, we have investigated the effect of a HFE on hysteresis and on synchronization area in a forced van der Pol–
Duffing oscillator near the 3:1 subharmonic resonance. We have performed an analytical approach based on averaging pro- cedure and multiple scales technique and derived successively a slow dynamic, its slow flow and the corresponding ‘‘slow
1034 A. Fahsi, M. Belhaq / Chaos, Solitons and Fractals 42 (2009) 1031–1036
slow” flow. Analysis of equilibria and periodic solution of these flows provides analytical expressions of the backbone curve and of the quasiperiodic response of the slow dynamic. The results shown that adding a HFE causes the backbone curve to shift and to bend from softening to hardening or vice versa. The main conclusion is that a fast excitation can be chosen so as to completely eliminate the hysteresis loop close to 3:1 resonance. It is also shown that in the softening case, the 3:1 sub- harmonic resonance may occur in a small range of the forcing frequency, whereas in the hardening one the 3:1 resonance area is large. The results conclude that a large hysteresis loop producing a jump phenomenon can take place near the 3:1 resonance, and in the hardening case, this resonance is significant and cannot be ignored in practical applications.
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2.7 2.75 2.8 2.85 2.9 2.95 3
0 0.5 1 1.5 2
ω
r−amplitude
Ω=0
(a)
E
QP
QP
2.45 2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85 0
0.5 1 1.5 2
ω
r−amplitude
Ω=40
(b)
Fig. 3.Entrainment area (E) and modulation amplitude of slow flow limit cycle (QP).
0 10 20 30 40 50
0 0.2 0.4 0.6 0.8 1
Ω
Hysteresis area
Ωc
Fig. 4.Variation of the hysteresis area versus the frequencyXnear 3:1 subharmonic resonance.
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